VYGOTSKY AND THE ZPD: HAVE WE GOT IT RIGHT?

This article was downloaded by: [Laurentian University] On: 10 October 2014, At: 05:53 Publisher: Routledge Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Research in Mathematics Education Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/rrme20 VYGOTSKY AND THE ZPD: HAVE WE GOT IT RIGHT? Stuart Rowlands a a Centre for Teaching Mathematics , University of Plymouth , Published online: 14 Apr 2008. To cite this article: Stuart Rowlands (2003) VYGOTSKY AND THE ZPD: HAVE WE GOT IT RIGHT?, Research in Mathematics Education, 5:1, 155-170, DOI: 10.1080/14794800008520120 To link to this article: http://dx.doi.org/10.1080/14794800008520120 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. 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Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and- conditions D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 http://www.tandfonline.com/page/terms-and-conditions http://www.tandfonline.com/page/terms-and-conditions VYGOTSKY AND THE ZPD: HAVE WE GOT IT RIGHT? Stuart Rowlands Centre for Teaching Mathematics, University of Plymouth There is a disparity between the historical Vygotsky and the diversity of 'our Vygotsky '. This disparity is complex because it not only involves the interpretation of text but also mistranslation, construction of meaning and the legitimisation of current trends. This article attempts to throw light on the historical Vygotsky by unpacking this disparity with the ZPD in particular and argues that the ZPD should be seen as part and parcel of a scientiJic method in the quest to change psychology into an abstract theoretical framework, similar in structure to Marx's Capital. In relation to Vygotsky, the issues of interpretation, translation, the logic of Capital and the 'scientzjically correct method' are discussed. In particular, the ZPD is discussed in the context of teaching mathematics as a formal, academic discipline. INTRODUCTION Since the beginning of the Nineties the prominence of Vygotsky has been phenomenal. Judging by the number of references to Vygotsky over this period, mathematics education research is no exception [I]. A hitherto relatively unknown Soviet psychologist, who wrote in the aftermath of revolution and within the context of building a socialist state, has become a posthumous 'authority' in the current educational scene. There is, however, a big irony here. Although Vygotsky wrote according to the needs and conditions of the time, his writings are not specific to those needs and conditions but can be shown to have a universality that is applicable to psychology and education. The irony is that education research seems to have missed that universality. It seems as if Vygotsky has become the icon for the 'social'. The popular notion in the Eighties of the lone organism adapting as it makes sense of the world has been overshadowed by such notions as 'situated cognition', 'activity theory', 'community of practice' and 'peer group collaboration' - all of which has the stamp of Vygotsky. He has become so much the symbol for these various viewpoints that emphasise the social that many of his ideas seem to have been overlooked: Fame is a socially constructed entity which functions for the purposes of the constructors, rather than for the designated bearers of that role themselves. A fitting proof of the societal construction of Vygotsky's stature is the list of ideas that the fascinated public has been persistently overlooking in the discourse about Vygotsky. (van der Veer and Valsiner, 1994, p.5) Van der Veer and Valsiner (1993) have stated that the references to the 'genius' of Vygotsky are "a good means of advertising but perhaps not conducive to an understanding of the content and implications of the ideas of the 'genius'. " (p.1). This implies a disparity between 'our Vygotsky' [2] and what Vygotsky was writing about. The aim of this article is to shed light on the universality of Vygotsky and D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 argues that this universality may be nothing less than the attempt to build psychology according to the 'scientifically correct method7 of Marx. To place the ZPD in context, this article argues that the ZPD should be seen as part of the quest to develop psychology into a science. As a Marxist, Vygotsky would have believed that, subject to extraneous factors such as the reader's development or his or her relation to the class struggle, he could convey what he wanted to mean in Russian. Of course, as in any language, there are nuances in Russian that cannot be translated directly into English - so unless we can read Russian (and this applies to the author, who unfortunately is ignorant of Russian), we cannot know exactly what Vygotsky wanted to mean. However, the point is, it would seem unlikely for Vygotsky to write purely to stimulate or legitimise our personal worldview: he not only wrote to convey his ideas but to change ours. But what were his ideas? One way to begin answering this question is to unpack this disparity. Ultimately, the answer lies in his text, but this is no simple matter. For one thing, an interpretation of Vygotsky may not be an interpretation of text but a point of view legitimised by reference to Vygotsky. If that point of view is to be challenged on the grounds that it is not a fair representation, then reference ought to be made to the relevant text. However, what text is deemed relevant may depend on yet another point of view. One way out of this 'herrneneutic circle' is to underpin Vygotsky with reference to Marx and Engels. Here the problem doesn't go away, especially as there are probably more versions of Marx than there are Vygotsky, but placing one within the context of the other may shed more light on the historical Vygotsky (and perhaps the historical Marx), especially if anchored in text. The next section claims that it makes sense to speak of a 'final version' of Vygotsky by attempting to reveal the disparity between the 'historical Vygotsky' and 'our Vygotsky'. With reference to an example taken from mathematics education research, the complex issue of interpretation is discussed. This section also discusses the translation of Vygotsky's two popular works and how this may have affected our understanding of Vygotsky. The subsequent section discusses the ZPD primarily in the context of the 'scientifically correct method7 of Marx and within Vygotsky's quest to build psychology according to this method. To locate what Vygotsky meant by science, this section includes a brief description of the logical form of Capital and how this compares with mechanics and the psychology Vygotsky was attempting to build. The following section is the conclusion, which argues the need to refer to his collected works and begin a new discourse. THE DISPARITY BETWEEN THE HISTORICAL VYGOTSKY AND 'OUR VYGOTSKY' Can there be a final version of Vygotsky, or can there only be 'our Vygotsky' consisting of a myriad of interpretations that cannot be compared to a final version? Can we accommodate an historical Vygotsky or can he only be assimilated into our D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotsky and the ZPD: Have we got it right? present schema of things? On the one hand it makes sense to say that he wrote so as to convey what he meant. It makes sense to say this because, otherwise, all versions of Vygotsky - 'our Vygotsky' - would collapse into a uniform relativism in which each version stands in equal relation to Vygotsky. For example, someone could claim Vygotsky is saying x, that the ZPD is concerned with normative skills: The upper level of the ZPD is normative. In one society, the skill of rowing a kayak around the age of 6 years becomes an expectation for children; in another society, it is riding a bike. (Hedegaard, 1998, p. 1 19). Someone else could claim that he is saying not-x, that the ZPD is not concerned with normative skills: The kind of instruction Vygotsky had in mind was not concerned with "specialized, technical skills such as typing or bicycle riding, that is, skills that have no essential impact on development" (Vygotsky, 1934, p.222), but rather had as its goal "all-round development", such as instruction in formal, academic disciplines, each of which has a sphere "in which the impact of instruction on development is accomplished and fulfilled" (ibid.).' Wertsch (1985, p.71) and neither of them, in the eyes of many, would be right or wrong. However, from the two passages it would be reasonable to contrast what Vygotsky had in mind to what Hedegaard has in mind: the first passage reveals the way Vygotsky has been assimilated, the second passage is anchored in text. This is not so much a disparity between different interpretations of Vygotsky's text. Rather, this is more a disparity between different versions of Vygotsky. For example, Hedegaard places the ZPD in terms of kayak rowing which would be contrary to what Wertsch has described as Vygotsky's evolutionary view of universal human rationality. According to Wertsch: 'Primitive thinking' in general differs from modern forms in that the former does not rely on abstract concepts. Such abstract concepts are viewed as emerging at a later historical point. One of the results of this formulation is that what we would today call cross- cultural differences were for Vygotsky and his colleagues 'cross-historical' in nature (see Wertsch, 1985). (Wertsch, 1996, p.27). Now contrast this passage from Wertsch with the following: The underlying assumption behind the concept [ZPD] is that psychological development and instruction are socially embedded; to understand them one must analyze the surrounding society and its social relations. (Hedegaard, 1996, p. 17 1, emphasis added) Here we have a difference in what may be described as outlook, perspective or 'world view' (weltanschuung). Both world-views are purported to be Vygotsky's [3]. On the other hand, how can we ever know what meaning Vygotsky had intended? A short and obvious answer is to read his works, but there are two problems here. One involves interpretation and the other translation. Let us begin with the first. D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 Interpretation The problem of interpretation is not only a problem of text, but of imposed meaning. Many articles on Vygotsky are not so much scholarly accounts of his writing, rather, they are versions of Vygotsky within contemporary education. Consider the following example from mathematics education. With reference to the 'transmission model of learning', Crawford states: In general, researchers have interpreted Vygotsky's concept (of the ZPD) in non dialectical terms that fit more closely with the assumptions of a transmission model of learning, in which an expert 'teaches' a novice, that underlie the social organisation of existing institutionalised settings. . . . (p.43). Vygotsky's view of human learning, and the developmental experience of being and acting in a cultural context, challenges many of the epistemological beliefs and assumptions underlying educational practice. In particular, it challenges traditional views of mathematics as value free, objective and divorcedfiom everyday personal concerns. . . . (p.44, emphasis added). Crawford is claiming that Vygotsky's view of human learning challenges both a 'transmission' interpretation of the ZPD and the view that mathematics is value free, objective and divorced from everyday personal concerns. Indeed, many writers such as Hatano (1993) and (Litowitz, 1993) make the first claim, but it is not at all clear how "researchers in general" have interpreted Vygotsky's concept of the ZPD in terms of a transmission model of learning [4]. Nor is it clear what a 'transmission model' means. Presumably, in the broad context given, it is teaching in a classroom. The second claim, however, becomes directly a question of text. Compare the second claim with the following by Vygotsky: Absolute correctness is achieved only beyond natural language, in mathematics. Our daily speech continually fluctuates between the ideals of mathematical and of imaginative harmony. (Vygotsky, 1962, p. 127) Vygotsky speaks of mathematics as an ideal and is it difficult to see how this challenges mathematics as value free, objective and divorced from everyday concerns. Now consider the following passage from Vygotsky: So this, generally, means that if we are dealing with a situation where this ideal form is not present in the environment, and what we have is interaction between several rudimentary forms, the resulting development has an extremely limited, reduced and impoverished character (p.350). . . . . Imagine a child who will develop his concept of numbers, his arithmetical thinking, only among other children, who will be left to his own devices in an environment where no developed form of arithmetical thinking exists, rather than in school or in kindergarten, i.e., without any interaction with the ideal form of adults. What do you think, will these children get far in developing their arithmetical thinking? None of them will, not even the mathematically gifted ones among them. Their development will remain extremely limited and very narrow in scope (Vygotsky, 1994, p.35 1). D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotsky and the ZPD: Have we got kt right? Taken together, both passages from Vygotslcy simultaneously contradict both claims by Crawford. Vygotsky speaks of mathematics as an ideal and not something that should be context-specific or something that can be developed simply with the child in social activity with others [5]. He is arguing that learning cannot take place unless this 'ideal form7 exists amongst adults. There are two issues here. Firstly, Vygotsky's view of human learning does not challenge traditional views of mathematics, rather, Vygotsky has been 'invoked' as an authority to legitimise the challenge against traditional views of mathematics. Secondly, it is not Vygotsky but contemporary socioculturists who challenge the teaching of formal academic subjects in the classroom setting. This challenge comes in the form of a 'social' version of the ZPD. For example, Moll and Whitmore (1993) uses the ZPD as a concept for the 'child-in- social-activity' that emphasises sociocultural conditions; McNamee (1990) refers to the 'community7s ZPD' and Moll and Greenberg (1990) discusses the ZPD within the context of the 'funds of knowledge' of different class family backgrounds that ought to be brought into the classroom [6]. It must be stated at the outset that Crawford's two claims do not reflect on her integrity and scholarship. Although Crawford is expressing a version of Vygotsky that can be shown to be contrary to the historical Vygotsky, nevertheless this version is well established in the literature. There has been a diversity of interpretation with respect to Vygotsky's ideas and it is conceivable that, for many, this diversity of interpretation should be celebrated: "after all, for Vygotsky and his collaborators to have stimulated such diverse contemporary ideas in education can only be a good thing!" Whether or not a diversity of ideas is a good thing may depend on those ideas and their implications. The point, however, is that this disparity between Vygotsky and 'our Vygotsky' seems more to do with the imposition of meaning, or rather the assimilation into a 'world view7, than it has with simply interpreting his words. The issue is complex, and that may be because it is not only an issue of assimilation and interpretation but also the way Vygotsky has been translated. Translation Gillen (2000) argues that the translations of Thought and Language (Vygotsky, 1962, and the 1986 edition) and Mind in Society (Vygotsky, 1978) are influenced by the ideologies of the translators, influencing the way Vygotsky's theories have been received. For example, the 1962 edition of nought and Language was a translation written in the aftermath of the McCarthy witch-hunts with no fundamental changes made in the 1986 translation. A comparison with the English edition of Thinking and Speech (Vygotsky, 1987), which is meant to be the same book, reveals the distortion and ambiguities of the English edition of Thought and Language in which references to Marx and Engels have been downplayed or omitted. Gillen (1999) provides a convincing case for the mistranslation of Vygotsky7s two most widely read books: Thought and Language and Mind in Society. The way a work has been translated has an effect on the way that work is publicly interpreted. In education, the prominence of Vygotsky is partly due to poor translation which has led to what Gillen (2000) D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 calls a 'facilitation of false coherence', that is, the 'temptation of finding an overly simple sense of coherence in Vygotsky's work7 with the ZPD in particular (p.185). Of course, the quality of translation is not a sufficient cause for the popularity of the work - there has to be other factors involved in the creation of what could be referred to as the 'Vygotsky Phenomenon7. Gillen gives two: Vygotsky has been used as an authority in support of current trends because the political and historical distance of Vygotsky has made it convenient to divorce his ideas from the context and debates that gaverise to those ideas. Secondly, Vygotsky appeals to those who have a concern with social/moral/political issues and education. This implies a reading of his collected works which apparently is a more faithful translation. If his collected works have not been mistranslated, then we should begin a new discourse with reference to his collected works (or at least the 'Vygotsky Reader', an anthology of the collected works). Studying the collected works of Vygotsky is necessary if we are to understand the historical Vygotsky, but it may not be sufficient. It may also be necessary to locate Vygotsky's work within the material conditions of the Soviet Union at the time and try to understand the Bolsheviks7 plan to raise the educational level s f the masses so as to overcome those conditions (for example, volunteers responding to the call to raise literacy, which puts into context Vygotsky's 'more competent peers.' Rowlands, 1999, 2000). This is a complex issue involving the notion of historical materialism [7] but, unfortunately, there is not the space to discuss the historical context. Unfortunately, because the historical context would indicate the political and historical distance between 'our Vygotsky' and the Vygotsky situated in the context and debates of the time. Nevertheless, it may be possible to locate his central ideas within a universal context. The next section is an attempt to begin a new discourse on those ideas by first discussing the ZPD as method and then locating this method within the context of building psychology into a science. However, this is more a suggestion so as to provoke debate than a claim on the 'final version7 of Vygotsky. THE ZPD AS CONCEPT AND METHOD IN THE ATTEMPT TO FQRMULATE AN ABSTRACT THEORETICAL FRAMEWORK Introduction Vygotsky defines the Zone of Proximal Development as 'the distance between the actual developmental level as determined by independent problem solving and the level of potential development as determined through problem solving under adult guidance or in collaboration with more capable peers' (Vygotsky, 1978, p.86). The ZPD is thus a relational concept between the two concepts actual development and potential development, Vygotsky (1978, 1987) speaks of that relation as one of 'instruction' or mediation. This 'instruction7 (not to be taken as the 'transmission model' of didactic teaching) awakens those higher psychological functions that are in a process of maturation. The implication is that without instruction, developmental D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotsky and the ZPD: Have we got it right? growth will become stunted and the child will remain at her actual level of development. Vygotsky (1987) saw the ZPD as an essential method in understanding the dynamic between instruction and development and used the ZPD as a concept central to his criticism of academic readiness. For example: Research indicates that the zone of proximal development has more significance for the dynamics of intellectual development and for the success of instruction than does the actual level of development. (Vygotsky, 1987, p.209) A central element to Vygotsky's work was the relation between scientific concepts and spontaneous concepts and in Thinking and Speech the ZPD is central to this relation, e.g. "Scientific concepts restructure and raise spontaneous concepts to a higher level, forming their zone of proximal development. " (Vygotsky, 1987, p.220). PI It is difficult to make the distinction between the ZPD as concept and method. On the one hand, Vygotsky speaks of the ZPD in terms of, for example, pretence play (Vygotsky, 1978) and the relation between scientific and spontaneous concepts (Vygotsky, 1987). On the other hand, he speaks of the ZPD as a tool through which the internal course of development can be understood (Vygotsky, 1978). Broadly speaking, the ZPD is instruction, play, scientific concept or whatever is ahead of development but leads that development. The ZPD as concept seems to also imply method, because to understand development (which is presumably the whole point of the ZPD) we have to see how the child responds to the mediation, but in doing so, we interfere with the process of development. Lipman eloquently explains this: In contrast to Piaget, who seemed to ignore completely the effect his observational and interviewing techniques may have had on the mental behaviour of the children he is describing, Vygotsky stressed the interconnectedness of the mode of experimental or pedagogical intervention and the resultant development of the mind of the child. This reminded me of the Heisenberg's indeterminacy principle - that there is no way a particle can be observed without disturbing it. Our knowledge of things can be only our knowledge of how they act as a result of our observing and experimenting with them. For if the child's mind "as it truly is" cannot be that mind at rest and undisturbed, then we must face up to accept the dramatic alternative: The child's mind is likewise defined by how it acts when probed and stimulated by countless forms of intervention. If, then, the child's mind realises itself as a result of a variety pedagogical interventions, then teaching and mental development cannot be served from one another. To paraphrase Vygotsky, the only good teaching is that which stays ahead of development and draws it up behind. Moreover, if the child's cognitive performance is a function of the teacher's pedagogical performance, then we must find ways of greatly expanding the teacher's capacity for challenging the child to respond. (Lipman, 1996, p. xiii) D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 The following is an attempt to place the ZPD within the fkamework of Vygotsky the Marxist, who had definite views on method and its relation to building a psychology that has a structure resembling Capital. Method According to Newman and Holzman: Traditional Science - including radical, ecologically valid science - sets up experimental situations that replicate real life and uses them to describe what is, in the Marxian sense, alienated reality. The Vygotskian enterprise, as we see it, is to create zones of proximal development - environments where people can perform life - and in so doing transform alienated reality. The difference could not be greater. (Newman and Holzman, 1993, p.29, italics added). Newman and Holman have located the ZPD in 'performing life' outside the experimental situation and this, it shall be argued, is contrary to the historical Vygotsky. Below is an attempt to locate the ZPD within the context of building an abstract theoretical framework. Meanwhile, the authors argue that the laboratory setting is contrary to Vygotsky's perspective because it distorts 'life-as-lived' and does not transform life as a 'revolutionary' activity. However, according to van der Veer and Valsiner (1993), for Vygotsky, the greater the artificial conditions the closer we are to understanding the psychological processes scientifically. Van der Veer and Valsiner state that: In his [Vygotsky's] view science was based on the reconstruction and interpretation of indirectly given phenomena, and in this respect, he saw no fundamental differences between the natural and social sciences and the study of history. Referring to Max Planck and Engels, Vygotsky argued that all of these sciences transcend the directly visible by making use of instruments and making inferences about the unknown. (van der Veer and Valsiner, 1993, p. 148) Science is based on the reconstruction of indirectly given phenomena. That reconstruction is experiment and implies artificially creating the conditions according to some theory to show that nature can be described by that theory. Vygotsky based his method on Marx's 'Thesis on Feuerbach' (Wertsch, 1985; Rosa and Montero, 1990) which was Marx's attempt to resolve the fundamental epistemological problem concerning the relationship between thought and reality that the great philosophical schools failed to satisfactorily answer. Marx's solution to the problem is expressed in his well-known statement: The philosophers have only interpreted the world, in various ways; the point, however, is to change it (XI Thesis on Feuerbach. For explanation, see Suchting, 1986). Quoted in the forefront of Vygotsky's (1978) Mind in Society is a quote from Engels' Dialectics of Nature: "It is precisely the alteration of nature by men, not nature as such, which is the most essential and immediate basis of human thought." This epistemological statement implies method, although probably due to bad translation, this method is not D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotsky and the ZPD: Have we got it right? apparent in Mind in Society. However, according to Wertsch, Vygotsky called his method 'experimental-developmental': which calls for an experimenter to intervene in some developmental process in order to observe how such intervention changes it. Again the primary motivation for doing this is to observe genetic processes: "Our method may be called experimental-developmental in the sense that it artificially provokes or creates a process of psychological development" (Vygotsky, 1978, p. 6 1-62). (Wertsch, 1985, p. 19). For Vygotsky, an artificial combination of conditions can be created to reveal the action of some speczfied law (e.g. the law of cognitive development. See Vygotsky, 1987) in its clearest form (van der Veer and Valsiner, 1993), and this is consistent with Marx's 'scientifically correct method' of 'ascending from the abstract to the concrete' : This process, stated paradoxically by Marx as 'ascending from the abstract to the concrete', is claimed to be 'obviously the scientifically correct method'. There is good evidence for Marx in affirming this. A mathematical science requires analysis and abstraction; it requires clearly defined theoretical objects (p.105). . . . The totality of science as it appears in the head is a 'product of a thinking head'. Science does not begin with real objects in the world but with intellectually constructed objects, with conceptions. (Matthews, 1980, p. 106). In other words, in theoretical practice we have first the (theoretical) idea then the realisation of the idea in practice (experiment). This is consistent with Chalmers' (1982) and Toulmin's (1969) argument that the scientist does not perform an experiment without some theoretical point in mind and that the experiment is structured according to the theory. Hestenes (1992) goes so far as to define experimental activity as 'model deployment games' such that a Newtonian model, structured by the laws of motion, is prior to any experimental activity. The point is, the intervention in the 'experimental-developmental' method is structured according to the 'ideal' (e.g. according to the nature of the concepts to be acquired by the child or the task to be completed) but 'in accordance' with the response of the child. For example, based on this method is the principle of 'double stimulation': a problem is set and the child is observed in solving it. Then a ready method is set and the child is observed in applying it. An 'intermediate function', or 'executive instrument', is constructed from the two observations, thus enabling the researcher to mediate between the task and its fulfilment (see Luria, Sakharov, Vygotsky and Luria in The Vygotsky Reader, Vygotsky, 1994; and 'Experimental Study of Concept Development' in Vygotsky 1987). The ZPD, I would argue, should best be seen in this light. The ZPD treats cognitive development as a development in process and change rather than as an end-product established as a set of discrete levels. For example, (taken from Rowlands, 1999, 2000, and Rowlands et al., 1996) if a child can successfully complete a task unaided, then prior knowledge of the abilities required to complete the task would merely D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 enable us to say what abilities the child has - 'merely' because we would be looking at the child's abilities that have already matured in the child - we would be looking at a 'snap-shot' of the maturation process as an end-product. To understand the maturation process as a process then we would have to facilitate the child's completion of a task that the child cannot do unaided. How a child responds to the mediation in completing a task enables us to explain the abilities of the child as they mature, rather than simply describe the abilities that have already developed. The mediation is structured by the task (the ideal) but is in accordance with the response of the child and should therefore lead the necessary psychological development towards accomplishment. In the teaching of formal, academic concepts, the ZPD is structured according to the 'target concept9 - but what questions to ask, hints to use, elaboration and demonstrations to make are part determined by how the class responds to the mediation of the target concept. The ZPD is thus a diagnostic tool as well as a 'method' for attaining the target concept. For example, the target concept may be a general method for solving linear equations (that may lie within the potential development of the class) and the class may be able to solve simple equations of the fonn x + a = b by guessing values for x (actual development). The use of prompts, props and hints (e.g. analogous metaphors such as mobiles or a scale-balance), questions (eg. "If I subtract 3 from the LHS, what must I do to the RHS if LHS = RHS?) and examples for the class to 'imitate' should provoke a cognitive response. The response reveals the understanding of the class and is also an indication of what steps can be taken to arrive at the general method of solving linear equations, bearing in mind that the response is part of the dynamic created by the mediation and does not reflect their initial cognitive state. Whether it is classroom teaching, one-to-one tuition or within the context of building an abstract theoretical framework, the ZPD should be seen as part of Vygotsky's 'scientifically correct method' of ascending from the abstract to the concrete. That abstract is the 'ideal', which, in the broader context, is an abstract theoretical framework. The next subsection discusses Vygotsky's attempt to transform psychology into a science and examines the logic of the theoretical framework that, for Vygotsky, constitutes a science. The Universality of Vygotsky I want to find out how science has to be built, to approach the study of the mind having learned the whole of Marx's method. In order to create such an enabling theory-method in the generally accepted scientific manner, it is necessary to discover the essence of the given area of phenomena, the laws according to which they change, their qualitative and quantitative characteristics, their causes. It is necessary to formulate the categories and concepts that are specifically relevant to them - in other words, to create one's own Capital (Vygotsky, 1978, p.8, author's italics). There is evidence to suggest that Vygotsky's quest was to transform psychology into a science, and by science he meant a theoretical framework that has the same logical structure as Marx's Capital: an axiom that defines the central unit of analysis D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotslcy and the ZPD: Have we got it right? (commodity value) from which a hierarchy of concepts are all well-defined by this central unit (surplus value, value of labour-power, constant and variable capital, rate of surplus value, rate of profit). For example, the law of the tendency for the rate of profit to fall, central to Marx's theory of capitalist crisis (volume 3), is derived from the law of commodity value (volume 1, chapter 1). According to Wertsch: In an unpublished notebook Vygotsky (1978) observed that: "the whole of Capital is written according to the following method: Mam analyzes a single living cell of capitalist society - for example,' the nature of value. Within this cell he discovers the structure of the entire system and all of its economic institutions. He says that to a layman this analysis may seem a murky tangle of tiny details. Indeed, there may be tiny details, but they are exactly those which are essential to 'microanatomy'. Anyone who could discover what a 'psychological' cell is - the mechanism producing even a single response - would thereby find the key to psychology as a whole. (Cole and Scribner, 1978, p.8)". Thus in order to pursue his study of human consciousness, Vygotsky viewed his task as one ofidentzfiing an investigable microcosm. (Wertsch, 1985, p. 193, emphasis added) [9] Vygotsky was well aware of the logical structure of Capital (see Rowlands, 2000) which is identical to that of Newtonian mechanics: the central unit being force, implicitly well-defined by 3 axioms (Hestenes, 1987, 1992) from which we have a hierarchical system of concepts (force, work, energy, power) [lo]. For example, the law of conservation of momentum, is derived from the three laws of motion (an analysis of the logical structure of mechanics can be found in Hestenes, 1987, 1992). Indeed, Capital is the only example of a social science that has the same logical structure as a domain ofphysics (mechanics). There is evidence and every reason to suppose Vygotsky wanted psychology to be the second [ I l l . If this was Vygotsky's quest, then the ZPD, in the light of method, can be viewed as part and parcel of building an abstract theoretical framework. His law of cognitive development and the dynamic between scientific and spontaneous concepts (Vygotsky, 1987) suggests his task was well underway. Of course, much of this is still speculation as Vygotsky only spoke of the ZPD and the logical form of Capital just before his death. However, the evidence suggests that his struggle to find the central-unit in psychology (the 'psychological cell') and to build a framework structured by this central unit was not a priori but actively invoked method to which the ZPD was not only part but seemed to be a culmination. CONCLUSION: THE NEED TO BEGIN A NEW DISCOURSE AND TO READ HIS COLLECTED WORKS The ideas of a thinker may prompt new ideas that are different to what the thinker intended. However, that difference has to be made explicit. There is a sense in which someone can be taken out of context and there is nothing wrong in this, provided this change of context has been made apparent. For example, Marx stated that he turned Hegel 'on his head' and fially acknowledged Hegel's ideas for making his own possible - but he never called himself an Hegelian. If we take Vygotsky out of D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 context then we should be clear that this is what we are doing and should not call ourselves Vygotskian for doing so. This may require studying Vygotsky in the spirit of what he was attempting to say. The essential point is that, if Vygotsky's work is to have any relevance or importance in education (over and above the plethora of idiosyncratic interpretations) then we have to start again with a new discourse, put aside our preconceptions and seriously try to understand his ideas. The reaction to this may be of indifference: "why should anyone care what his ideas actually were, isn't this all a bit 'academic'?" My response is that by seriously attempting to accommodate someone's ideas, rather than merely assimilating them, our ideas can collectively undergo a process of development: a development in new ideas hitherto unthought rather than our present ideas in new wine bottles wrongly labelled Vygotskian. That development may eventually lead to ideas that are contrary to Vygotsky's. Indeed, we may eventually and consciously turn Vygotsky on his head, but this will not be possible if Vygotsky is merely assimilated into our present schema of things. And the relevance to mathematics education? Wertsch (1996) has described Vygotsky as an Enlightenment Abstract Rationalist, and this may be an accurate description concerning his theory of scientific and spontaneous concepts. Educationally, Vygotsky is primarily concerned with how children internalise scientific ('academic', or 'schooled') concepts which arguably has a direct relevance to mathematics teaching. What we all make of this still remains to be seen [12], but it may be necessary to begin studying his collected works to assess what contribution Vygotsky can make to mathematics education and education in general. NOTES 1. For example, Chassapis (1 998/1999), Crawford (1996)' Jaworski (1994)' Jaworski (2002), Ernest (1991)' Ernest (1994), Gardiner (1998)' Hudson et a1 (1999), Lemerise (1993)' Nickson (2000), Wilson et a1 (1993). 2. An expression from Newrnan and Holzman (1993). 3. Vygotsky the Enlightenment Abstract Rationalist (Wertsch, 1996) and Vygotsky the socioculturist. 4. Unfortunately, no references have been forthcoming. What we have, apparently, is an assumption that any emphasis on the role of the teacher or the subject matter to be taught implies the transmission model of learning. What is normally understood as the transmission model, however, is the teacher transmitting knowledge and the pupil passively receiving it. 5. It must be noted that Vygotsky's 'ideal form' is not a Platonic Form as such (it is most unlikely that Vygotsky believed in the ontological existence of ideal forms. Ironically, this is probably true of Plato, who argued the existence of the Forms so as to emphasise the Forms as a heuristic distinct from particulars). Vygotsky's point is that children can only learn mathematics if the theoretical objects of D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Vygotsky and the ZPD: Have we got it right? mathematics (the 'ideal form' consisting of theoretical objects such as numbers, operations, points and triangles, etc) exists amongst adults (and conversely, without adults, the ideal form would not exist). 6. It seems as though many socioculturists have used the ZPD as if it was a metaphor for their awareness that 'it is not the consciousness of people that determines their being, but their social being that determines their consciousness' (Rowlands, 2000). 7. An in-depth materialist analysis of the Soviet Union can be found in Fiiredi (1986). 8. Although this relation between scientific and spontaneous concepts is discussed in Thought and Language (Vygotsky, 1962), there is no mention of the ZPD, even though Thinking and Speech and Thought and Language are meant to be the same book. 9. After many years of work and just before his death Vygotsky identified the central unit of analysis as emotional experience An emotional experience {perezhivanie) is a unit where, on the one hand, in an indivisible state, the environment is represented, i.e. that which is being experienced - an emotional experience (perezhivanie) is always related to something which is found outside the person - and on the other hand, what is represented is how I, myseE am experiencing this. . . (Vygotsky, 1994, p.342, author's emphasis). Had he lived (and survived the Stalinist purges without censorship) then there is every reason to suppose that he would have constructed a framework of concepts that are all well-defined by emotional experience. What concepts Vygotsky would have defined by emotional experience is a matter of speculation, but the method by which he would have searched for the appropriate concepts would have involved artificially creating the conditions so as to provoke emotional experience, and to see how emotional experience manifests under these conditions. 10.As stated by Wittgenstein: "Mechanics determines one form of description of the world by saying that all propositions used in the description of the world must be obtained in a given way from a given set of propositions - the axioms of mechanics. It thus supplies the bricks for building the edifice of science, and it says. 'Any building that you want to erect, whatever it may be, must somehow be constructed with these bricks, and with these alone'. " (Wittgenstein, 1974. proposition 6.341, p.68). 11.Vygotsky did refer to psychology as a science and referred to the programme to transform psychology into a science as "our science" (and to "create one's own Capital"), but he did make clear that a science is not the combination of 'elements' (such as Stimulus-Response) but rather units that are all defined by a single unit of analysis (the study of which he called 'microanatomy7). See Vygotsky (1 994). D ow nl oa de d by [ L au re nt ia n U ni ve rs ity ] at 0 5: 53 1 0 O ct ob er 2 01 4 Research in Mathematics Education Volume 5 12.There presently exists a sociocultural 'version7 of Vygotsky that argues scientific concepts ought to be contextualised with reference to spontaneous, everyday concepts. This version is criticised in Rowlands (2000). REFERENCES Chalmers, A.: 1982, Wkat is This Thing Called Science? Second edition. Milton Keynes: 0 . U. Press. 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